Steiner triple systems with high chromatic index

Darryn Bryant, Charles Colbourn, Daniel Horsley, Ian M. Wanless

Research output: Contribution to journalArticle

Abstract

It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv2(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.

Original languageEnglish (US)
Pages (from-to)2603-2611
Number of pages9
JournalSIAM Journal on Discrete Mathematics
Volume31
Issue number4
DOIs
StatePublished - Jan 1 2017

Fingerprint

Chromatic Index
Steiner Triple System
Exception
Disjoint
Integer

Keywords

  • Block coloring
  • Chromatic index
  • Parallel class
  • Steiner triple system

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Steiner triple systems with high chromatic index. / Bryant, Darryn; Colbourn, Charles; Horsley, Daniel; Wanless, Ian M.

In: SIAM Journal on Discrete Mathematics, Vol. 31, No. 4, 01.01.2017, p. 2603-2611.

Research output: Contribution to journalArticle

Bryant, Darryn ; Colbourn, Charles ; Horsley, Daniel ; Wanless, Ian M. / Steiner triple systems with high chromatic index. In: SIAM Journal on Discrete Mathematics. 2017 ; Vol. 31, No. 4. pp. 2603-2611.
@article{13672b01d764447ead9babcf5654538b,
title = "Steiner triple systems with high chromatic index",
abstract = "It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv2(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.",
keywords = "Block coloring, Chromatic index, Parallel class, Steiner triple system",
author = "Darryn Bryant and Charles Colbourn and Daniel Horsley and Wanless, {Ian M.}",
year = "2017",
month = "1",
day = "1",
doi = "10.1137/17M1114338",
language = "English (US)",
volume = "31",
pages = "2603--2611",
journal = "SIAM Journal on Discrete Mathematics",
issn = "0895-4801",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "4",

}

TY - JOUR

T1 - Steiner triple systems with high chromatic index

AU - Bryant, Darryn

AU - Colbourn, Charles

AU - Horsley, Daniel

AU - Wanless, Ian M.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv2(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.

AB - It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv2(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.

KW - Block coloring

KW - Chromatic index

KW - Parallel class

KW - Steiner triple system

UR - http://www.scopus.com/inward/record.url?scp=85040375717&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85040375717&partnerID=8YFLogxK

U2 - 10.1137/17M1114338

DO - 10.1137/17M1114338

M3 - Article

VL - 31

SP - 2603

EP - 2611

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 4

ER -